A single asynchronous M/M/s queuing model is formed by a series of asynchronous servers denoted by Si where i∈N, (i.e. 1, 2, 3, 4, 5, …) and a lane of passengers waiting for being checked. Asynchronous servers handle passengers at different times, (i.e., every server in this queue model is independent to other servers). When a passenger finished his screening by the server, another passenger from the lane will take over that position to keep the server operating without pausing.

Figure 1 shows a typical single M/M/s queuing system fully loaded with costumers. For each server, ti denotes the time interval between two travelers moving through the gate. As soon as this passenger finishes this step, he moves to the following examination step and next costumer will come to the server and repeat this process, while ti varies for distinct passengers. To model ti, we need to analyze its components first.

Figure 2 shows a typical passenger divides his personal belongings into 3

baskets for x-ray examination. Each basket takes tb_j seconds to prepare. To generalize t_i for each passenger with n baskets, we have:

t i = ∑ j = 1 n t b j

According to most airline regulations, the luggage allowance for economy-class passengers is small. Therefore, we assume for all economy-class passengers, they can only separate their items into smaller than 6 baskets.

We model the number of baskets per passenger by a Poisson process. In addition, the Poisson distribution we employed is slightly modified to fit the case. Since the Poisson distribution with λ = 1 starts from 0, we add 1 to every Poisson random number to account for the fact that there is no one with no basket for examination. Furthermore, we cut all Poisson random numbers above 4, since we add 1 to every Poisson random number and assume no passenger will have more than 5 baskets.

Figure 3 illustrates the histogram of our modified Poisson distribution, which suggests 1 or 2 baskets per passenger are the most likely case and 4 baskets case is comparatively rarer than other cases. The time tbj takes by basket j is modeled by χ2 distribution with 2 degree of freedom. The number of degree of freedom is empirical based on an experiment. To be specific, in real world, a fully-loaded single M/M/s queue with 4 servers can handle a maximum of 1000 passengers in 1 hour (3600 seconds) [4]. We run the model of a single 4 servers M/M/s queue which handles 1000 people in 100 times. The mean of time to process these people is 3590 seconds while the standard deviation is 25.8017 seconds. The Figure 4 also suggests the results are quite steady around 3600 seconds.

The multiple asynchronous M/M/s queue model is composed of n single M/M/s queue models. The only difference is that, multiple asynchronous M/M/s queue

model is built to account for the fact that queues may be far away from each other and passenger are not informed to which queue is the shortest.

Although Figure 5 shows an illustration of multiple asynchronous M/M/s queue with same queue length, in our simulation or real world, this rarely happens. Hence we choose to imitate passengers’ behavior on random basis. We use multinomial distribution to model their decisions, meaning each queue is equally likely for each passenger to join. The multinomial distribution model will be discussed in more details in Section 4.

In our earlier investigation, most security checkpoints open from 4 am to 10 pm (16 hours). The arrivals are modeled as two stages [9]. The first stage is generated by a Poisson distribution which simulates with sampling interval down to 15 minutes. There are clusters of people arriving within each quarter. We calibrate the size of these clusters to 400 people per cluster to fit real world data. Every simulation of this Hybrid model will deliver an average people arrival rate of around 38,000 passengers per day. In second stage we use multinomial distribution to reduce sampling interval to 1 second, with data carried out from stage 1, and we uniformly distribute these clusters of people per 15 minutes to several people per second. We make this intensive sampling in order to plug passenger flows into our queue system for test purposes. Figure 6 illustrates how people arrive in different measurements. Figure 6(a) adds passenger flows in every quarter into an hour basis.

The second model is implemented by Markov chain. We can plot the state diagram as Figure 7 and define transition matrix by M.

The first passage hitting time for each state is calculated empirically from real- world data.

A typical day starts from state 1 and each hour is dependent on stochastic. The probability to reach states onward decreases over time. Therefore we use each state to represent a randomly selected hour in a day. State 1 stands for least busy, i.e. a small number of passengers’ arrivals in this state. State 2 stands for less busy, i.e. a normal number of passengers’ arrivals within this state. State 3 is the busiest state, i.e. a massive number of passengers arriving in the state. How-

ever State 4 is more like an idle state since we design this state with a first passage hitting time of 15, so we somehow expected it to be the state at the end of the day.

In our simulation (Figure 8), the Markov chain model gives 42,750 passengers to departure in a day. Although state 4 tends to be the last hour of the day in this simulation, state 4 is not necessarily to be the case if we run it for a number of times. It can also illustrate the case when there are only a few passengers arriving at the airport in a day.

Both models illustrate large variance within a day. Markov chain model runs faster since it has fewer states, while hybrid Poisson model has much more states compared to the Markov chain model and closer to real situations.

Set express bypass Swift for those people with one light luggage. These people, who only have few wait-for-checking items, can quickly pass through the Swift pass. This not only saves their time but also reduces the length of the regular pass. We set the threshold as no more than 5 kg luggage, which is the typical weight for basic traveling items for a business trip, a laptop (2 kg), a wallet, a backpack (1 kg), etc. A scale can be set at the entrance of this Swift pass so that this will not influence people using the regular pass. Figure 9 illustrates how swift queuing system works.

Another way to reduce the queue length is to encourage more passengers to apply for Pre-Check. According to TSA, approximately 45% passengers enroll in a program called “Pre-Check for trusted travelers”. These passengers pay an additional $85 to receive a background check and enjoy a separate screening process with few modifications to save time for five years [10]. There is a survey showing that 97% of TSA passengers only wait 5 minutes or less. We can promote this service by increasing more price options such as $20 per year vs. $85 per five year, which will attract more potential passengers such as international students who will not stay for five years in the U.S.